scholarly journals On Perfect Packings in Dense Graphs

10.37236/3173 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Andrew Treglown
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Edge Density ◽  
Structural Result ◽  
Dense Graphs ◽  
Delta G ◽  
Vertex Disjoint ◽  
Density Threshold

We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. We consider various problems concerning perfect $H$-packings: Given $n, r , D \in \mathbb N$, we characterise the edge density threshold that ensures a perfect $K_r$-packing in any graph $G$ on $n$ vertices and with minimum degree $\delta (G) \geq D$. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect $H$-packing. Other related embedding problems are also considered. Indeed, we give a structural result concerning $K_r$-free graphs that satisfy a certain degree sequence condition.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

scholarly journals Tilings in Randomly Perturbed Dense Graphs

2018 ◽  
Vol 28 (2) ◽  
pp. 159-176 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
ANDREW TREGLOWN ◽  
ADAM ZSOLT WAGNER
Keyword(s):  
High Probability ◽  
Minimum Degree ◽  
Graph Model ◽  
Regularity Lemma ◽  
Arbitrary Graph ◽  
Random Graph Model ◽  
Dense Graphs ◽  
Szemeredi's Regularity Lemma ◽  
Vertex Disjoint ◽  
Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

scholarly journals The bandwidth theorem for locally dense graphs

2020 ◽  
Vol 8 ◽  
Author(s):  
Katherine Staden ◽  
Andrew Treglown
Keyword(s):  
Maximum Degree ◽  
Blow Up ◽  
Minimum Degree ◽  
Bounded Degree ◽  
Dense Graphs ◽  
Vertex Graph ◽  
Delta G

Abstract The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

scholarly journals On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Shih-Yan Chen ◽  
Shin-Shin Kao ◽  
Hsun Su
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Sequence Characterization ◽  
Hamiltonian Connected ◽  
Vertex Graph ◽  
Minimum Number ◽  
Hamiltonian Connectivity ◽  
International Audience ◽  
Delta G ◽  
Hamiltonian Properties

International audience Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called <i>$k$-vertex fault traceable</i>, <i>$k$-vertex fault Hamiltonian</i>, or <i>$k$-vertex fault Hamiltonian-connected</i> if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

scholarly journals The Bandwidth Theorem in sparse graphs

2020 ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Anusch Taraz
Keyword(s):  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Main Tool ◽  
Sparse Graphs ◽  
Spanning Subgraphs ◽  
Dense Graphs ◽  
Vertex Graph ◽  
First Results ◽  
Delta G

One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

10.37236/8986 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Joseph Hyde ◽  
Andrew Treglown
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Lower Degree ◽  
Fundamental Result ◽  
Additive Constant ◽  
Perfect Graph

A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect $H$-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Charles Delorme ◽  
Guillermo Pineda-Villavicencio
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Minimum Degree ◽  
Integer Coefficients ◽  
Disjoint Cycles ◽  
The Matrix ◽  
Odd Degree ◽  
Unexplored Area ◽  
Trivial Graph ◽  
Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

Sign in / Sign up

Export Citation Format

Share Document